The Vertex-Face Correspondence and the Elliptic 6j-symbols

Abstract

A new formula connecting the elliptic $6j$-symbols and the fusion of the vertex-face intertwining vectors is given. This is based on the identification of the $k$ fusion intertwining vectors with the change of base matrix elements from Sklyanin's standard base to Rosengren's natural base in the space of even theta functions of order $2k$. The new formula allows us to derive various properties of the elliptic $6j$-symbols, such as the addition formula, the biorthogonality property, the fusion formula and the Yang-Baxter relation. We also discuss a connection with the Sklyanin algebra based on the factorised formula for the $L$-operator.

23 pages

References in corpus (5)

• Elliptic algebra U_{q,p}(^sl_2): Drinfeld currents and vertex operators
• Free Field Construction for Correlation Functions of the Eight-Vertex Model
• An Elliptic Algebra $U_{q,p}(\widehat{s{l}_2})$ and the Fusion RSOS Model
• Bethe Ansatz for higher spin eight vertex models
• Yang-Baxter Equation for $A_{n-1}^{(1)}$ Broken ${\Bf Z}_N $ Models

Cited by in corpus (4)

• Elliptic Quantum Group U_{q,p}(\hat{sl}_2), Hopf Algebroid Structure and Elliptic Hypergeometric Series
• New elliptic solutions of the Yang-Baxter equation
• Q-operators for higher spin eight vertex models with an even number of sites
• Q-operators for higher spin eight vertex models with a rational anisotropy parameter